ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The categorical trace of a linear endomorphism is its matrix trace

Example

For a finite-dimensional vector space $V$ and a linear endomorphism $T : V \to V$ , the categorical trace of $J_{V} \circ T : V \to V^{* *}$ is the ordinary trace of $T$ .

Facts & Assumptions

Given: A finite-dimensional vector space $V$ and a linear endomorphism $T : V \to V$ .

[L1]

The map $J_{V} : V \to V^{* *}$ is the canonical comparison (The canonical evaluation map $J_{V} : V \to V^{* *}$ given by $J_{V} (v) (f) = f (v)$ ).

[L2]

The ordinary trace of $T$ is defined basis-independently by The basis-independent trace of an endomorphism of a finite-dimensional vector space, and the categorical trace applies to $J_{V} \circ T$ by The categorical trace of a morphism into the double dual.

Verification

technique · direct

1.1

The canonical map $J_{V} : V \to V^{* *}$ from The canonical evaluation map $J_{V} : V \to V^{* *}$ given by $J_{V} (v) (f) = f (v)$ is the standard pivotal comparison for finite-dimensional vector spaces, so $J_{V} \circ T$ is an input for the categorical trace of The categorical trace of a morphism into the double dual.

givenL1L2

2.1

In a basis $(v_{i})$ with dual basis $(v_{i}^{*})$ , write $T (v_{j}) = \sum_{i} a_{i j} v_{i}$ . Then ${Tr}_{L} (J_{V} T) = \sum_{i} J_{V} (T (v_{i})) (v_{i}^{*}) = \sum_{i} v_{i}^{*} (T (v_{i})) = \sum_{i} a_{i i} .$

step 1.1algebra

3.1

The right-hand side is exactly the ordinary trace of $T$ by The basis-independent trace of an endomorphism of a finite-dimensional vector space. So the categorical trace specializes to the matrix trace in finite-dimensional linear algebra.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 4.7.1 and Example 4.7.10 (standard reference, not scraped)